The most common and simplest way is:
If we want to show more thanks:
Thanks a lot!
Thank you so much!
Thanks a million!
Ich danke Ihnen!
I give you my thanks (formal)
Ich danke dir!
I thank you! (informal)
Gott sei Dank
Vielen Dank im voraus
Thank you in advance (typically used when asking for a favor)
We think that German is quite difficult and these 2 simple words are an good example of this.
"Vielen Dank" is accusative.
Surely you are asking yourself: Why is "vielen Dank" declined in accusative?
Because when we say "Vielen Dank", we are omitting part of of the phrase. The phrase really is:
Ich sage dir vielen Dank
I say many thanks to you
Where "many thanks" completes the function of the direct object which is accusative in German.
viel means "much/a lot" and it is an adjective in German, which, in this case, has to be declined with the strong declension.
Currently, the custom of declining "viel" in the singular with the strong declension has been lost:
Malaga hat viel Wasser
Malaga has a lot of water
But the expression is so traditional that this custom continues:
That is why it is declined as: "vielen"
"Dank" means "thanks" and, just like all German nouns, it has to be capitalized.
There's a reason for everything!
The typical response to "danke" is:
No problem/ You're welcome
This is a little weird because "bitte" also means "please"
Other more formal responses are:
No problem/you're welcome
Nichts zu danken!
No reason to thank me!